Non-Hermitian dynamics of vortices in a shear flow (Study of Turbulence Structure : Generation, Dynamics, Statistics and Application)

Non-Hermitian dynamics of

vortices

in

a

shear flow

東大新領域 吉田善章, 龍野智哉 (Zensho Yoshida, Tomoya Tatsuno)

Graduate School

_of

Frontier Sciences, University

_of

Tokyo

1

I

ntroduction

Non-0rthogonality of eigenfunctions (modes) is the determining characteristic of

non-Her-mitian systems, which brings about interactions among different modes. This resembles

the mode couplings in nonlinear systems, and hence, the diversity of transient behavior in

non-Hermitian system is rather rich.

Let $1\mathrm{i}\mathrm{S}$ consider

an

abstract autono

nous

evolution equation of the Schr\"odinger tvpe

$\{$

$\dot{\uparrow,}\partial_{t}u=llu$

$\iota x$(0) $=\iota\iota_{()}$

(1)

where $\mathcal{H}$ is acertain linear operator. When we can generate an exponential function

(propagator) $e^{-it\mathcal{H}}$, we

can

write thc solution of (1)

as

$u(t)=e^{-it\mathcal{H}}\iota\iota_{0}$

.

When $u\in \mathrm{C}$ and $\mathcal{H}$ $\in \mathrm{C}$, then $e^{-it?t}$ is nothing but the exponential function

$()\mathrm{f}$elementary

mathematics. For vectors $\mathrm{s}\iota$

$\in \mathrm{C}^{N}$ and alinear map_$lt$ : $\mathrm{C}^{N}arrow \mathrm{C}^{N}$,

we

_call define $e^{-it\mathcal{H}}\mathrm{b}.\mathrm{v}$

the standard power series

$c^{-i\mathrm{f}H}=$ $n-,1 \sum_{--}^{\infty}\frac{(-it\mathcal{H})^{n}}{n!}$, (2)

or

the Cauchy integral (inverse Laplace transform)

$e^{-it\mathcal{H}}= \frac{1}{2\pi i}\oint e^{-- it\lambda}(\lambda I-\mathcal{H})^{-1}d\lambda$. (3)

For $u$ in aHilbert space 1

$r$

, $\mathcal{H}$ is

an

operator in I’. For

some

different

$\mathrm{c}1\mathrm{a}\mathrm{s}‘ \mathrm{s}.\epsilon_{\iota}\backslash \mathrm{s}$of operators,

we

have theories to generate $\epsilon^{t}-it\mathcal{H}$. For bounded operators,

we

can

invoke the Dunford

integral that is similarto (3). Amost general theory _{of generating}

an

_{exponential function}

of the type $e^{tA}$ for positive $t$ (so-called semigroup theory) is due to Hille and Yosidafl].

$\mathrm{A}1\mathrm{t}_{)}\mathrm{h}\mathrm{o}1\iota \mathrm{g}\mathrm{h}$ this theory warrants the solvability of initial value problems for $\dot{\mathrm{e}}1$ wide class

of generators, understanding of the behavior of the solution is not simple. Indeed, the

exponential functions of matrices or operators are not necessarily “exponential” in the

conventional

sense.

The

von

Neumallll theory for Hermit ian (self-acjjoint) operators provides adeep $\mathrm{i}_{11\mathrm{S}},$$\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}$

into the structure of$e_{\backslash }^{-- it\mathcal{H}}$ which

invokes the spectral resolution of the generator$\mathcal{H}$ in term

数理解析研究所講究録 1226 巻 2001 年 239-250

ofacomplete set oforthogonal modes. The basic idea is that the $e^{-it\mathcal{H}}$ may be represented

as

asum

ofindependent harmonic oscillators, each of which is

an

eigenfunction of $H$ and

the corresponding eigenvalue (real number) gives the frequency of the oscillation. Unlike

the

case

of finite dimension

vector

space,

however,

the conventional

eigenfunctions

may

not

be complete to span the Hilbert space. The most essential generalization needed to study

an

infinite

dimension space

was

the introduction of continuous spectra that correspond to

singular eigenfunctions. The spectral resolution of$\mathcal{H}$ is, in general, given by

an

integral

over

the

spectra (an example will be given in

Sec.

3.1). The contribution to the $e^{-it?t}$

from

the continuous spectra brings about the “phase mixing” of oscillations with continuous

frequencies, resulting in various types of damping. Hence, the reality of the spectra of

an

Hermitian operator does not necessarily imply stationary (non-dumping) oscillations.

For alinear map in afinite dimension vector space, the spectral resolution yields the

Jordan canonical form, and the explicit representation of $e^{-it\mathcal{H}}$

can

be constructed using

the canonical form. It is well known that anilpotent yields secular behavior of the

corre-sponding generalized eigenvector. Therefore,

even

if

every

eigenvalue $\lambda_{j}$ is real, the $e^{-it\mathcal{H}}$

can

describe “instabilities”

(growth

of

oscillations).

In aHilbert space, however, such ageneral theory of spectral resolution is limited to

either compact operators

or

Hermitian operators. This paper is

an

attempt to obtain

a

spectral resolutionofanon-Hermitian operatorthat isnot included in the above mentioned

categories. This operator is related to

an

important physics problem (Sec. 2).

2

_{Non-Hermitian}

_{dynamics of}

_vortices

The vortex dynamics equation in $\mathrm{R}^{2}$ [the coordinates

are

denoted by ($x$,$y$)] reads

as a

Liouville equation

$\partial_{t}\Psi+\{H, \Psi\}=0$, (4)

where

1is

the vorticity, $H$ is the Hamiltonian (stream function) of

an

incompressibleflow

$v=(\partial_{y}H, -\partial_{x}H)^{t}$ that transports the vortices, and

$\{a, b\}=(\partial_{y}a)(\partial_{x}b)-(\partial_{x}a)(\partial_{y}b)=-\nabla a\mathrm{x}$$\nabla b\cdot\nabla z$

is the

Poisson bracket.

When the Hamiltonian $H$ depends

o

$\mathrm{n}$ $\Psi$, the evolution equation (4) is nonlinear. The

dynamics of $\Psi$

can

couple with other fields when they

are

included in

$H$. The simplest

example of nonlinearvortex dynamics is that of the Euler fluid (incompressibleideal flow),

where

$-\triangle H=\Psi$, ₍₅₎

or, denoting the

Green

operator of the $\mathrm{L}\mathrm{a}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{n}-\Delta$ by $\mathcal{G}$

$H=\mathcal{G}\Psi$

.

(6)

Let

us

linearize (4) with decomposing 1and $H$ into their ambient (denoted by subscript

0) and

fluctuation

parts:

$\Psi$ _{$=$ $\Psi_{0}+\psi$},

$H$ $=$ $H_{0}+h=\mathcal{G}\Psi_{0}+\mathcal{G}\psi$.

Neglecting the second-0rder terms, (4) reads

$\partial_{t}\psi+\{H_{0}, \psi\}-\vdash\{\triangle H_{0}, \mathcal{G}\psi\}=0$. (7)

In this PaPer,

we

consider one-dimensional problem with

$H_{0}=H_{0}(x)$.

Since the ambient Hamiltonian $H_{0}$ is independent of _$y$, the wavenumber in $y$ is agood

quantum number, and

we can

replace $\partial_{y}$ by $ik$. In what follows,

we assume

$k\neq 0$, and

normalize $k=1[2]$. We write

$v(x)=-\partial_{x}H_{0}(x)$,

to obtain the standard Rayleigh equation

$i\partial_{t}\psi=v(x)\psi+v’(x)\mathcal{G}\psi$. (8)

The Green operator $\mathcal{G}$ is represented by aconvolution integral

$( \mathcal{G}f)(x)=\int_{-\infty}^{+\infty}\frac{e^{-|x-\xi|}}{2}f(\xi)d\xi$. (9)

In what follows,

we

denote by $G(x, \xi)$ the Green function;

$G(x, \xi)=\frac{e^{-|x-\xi|}}{2}$. (10)

3Convection and oscillations

The generator of the vortex dynamics equation (8) consists of two terms, each of which

describes different mechanism of vortex motion. The first termon the right-hand sideof(8)

[originating from $\{H_{0}$, $\psi\}$ in (7)] represents the transport of the vorticity by the ambient

flow $v(x)$. An inhomogeneous (sheared) flow distorts vortices, and hence,

no

stationary

structure

can

persist in ashear flow $(v(x)\neq \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t})$. Such adynamics is described by

acontinuous spectrum (Sec. 3.1). On the other hand, the second term [originating from

$\{\triangle H_{0}, \mathcal{G}\psi\}$ in (7)$]$ describes the interaction between the perturbation and the ambient

field. When the ambient vorticity $\Psi_{0}=-\triangle H_{0}$ has aspatial gradient, aflow induced by

a

perturbation yields alocal change ofthe vorticity. This term, hence,

can

create perturbed

vortices from the ambient field.

In this section, we study the role of both terms by formal calculations. In what follows,

it is convenient to generalize (8) with replacing $v’(x)$ by acertain “real” function $w(x)$

that is independent of$v(x)$. With assuming $k\neq 0$,

we

consider

$i\partial_{t}\psi=v(x)\psi+\tau v(x)\mathcal{G}\psi$. (11)

The

case

when $w(x)–v’(x)$

recovers

the physically relevant equation (8)

3.1

Convection

–shear flow

transport

Here,

we

assume

$w(x)=0$ in (11) and consider

$i\partial_{t}\psi=v(x)\psi$ (12)

with a“continuous” real function $v(x)$, which reads

as

aSchr\"odinger equation with

a

Hamiltonian $v(x)$.

The formal eigenvalue and the corresponding eigenfunction of the generator of(12), with

setting

$v(x)\psi=\omega\psi$

(i.e., $\psi(t)=e^{-i\omega t}\psi$), is given by

$\omega$ $=v(\mu)$, $\iota[$) $=\delta(x-\mu)$, (13)

where $\mu$ is

an

arbitrary real number and

$\delta$ denotes the delta-measure. For theconvenience,

we

write

$(f(x), \delta(x-l^{l))=}\int_{-\infty}^{+\infty}f(x)\delta(x-\mu)dx=f(l^{l})$.

Aformal spectral resolution of the generator is written

as

$v(x)f(x)$ $=$ $\int_{-\infty}^{+\infty}v(\mu)(f, \delta(x-\mu))\delta(x-l^{l})d/\iota$ (14)

$=$ $\int_{-\infty}^{+\infty}v(_{l}\iota)f(\mu)\delta(x-\mu)d\mu$.

Rigorous mathematical representation of this “continuous spectrum” is given by the

spectral resolution of the coordinate operator:

$xf(x)= \int_{-\infty}^{+\infty}\mu dE(\mu)f(x)$, (13)

where $\{E(\mu);\mu\in \mathrm{R}\}$ is afamily ofprojectors (resolution of the identity) defined by

$E(\mu)f(x)=\{$ $f(x)$ for $x\leq\mu$

0for $x>\mu$ (16)

The projector $E(\mu)$ gives aresolution of the identity:

$I= \int_{-\infty}^{+\infty}$ dE(\mu ). (17)

Using this representation ofthe coordinate operator,

we

can

write

$v(x)f(x)= \int_{-\infty}^{+\infty}v(l^{l})dE(\mu)f(x)$

.

(18)

The solution of (12) with initial condition $\psi(x, 0)=\psi_{0}(x)$ is given by

$\psi(x, t)=\int_{-\infty}^{+\infty}e^{-:tv(\mu)}dE(\mu)\psi_{0}(x)=e^{-:tv(x)}\psi_{0}(x)$ . (19)

242

3,2

_{Chandrasekhar}

model of

surface-waves

Non-Hermitian property stems from thesecondterm

on

the right-hand side of (11), because

the multiplicationof$w(x)$ andthe integral operator(; does notcommute. As

we

have noted,

this term represents the interaction between the perturbed flow and the ambient vorticity.

Physically, the non-Hermitian property implies the

non-conservation

ofthe ”energy” of the

vorticity, i.e., the enstrophy $\mathrm{J}^{\cdot}|\psi|^{2}dx$. We also remark that the original nonlinear system

(4)

conserves

the enstrophy, as well as all “Casimirs” $\int f(\Psi)dx(f$ is an arbitrary smooth

function). The non-conservation of the enstrophy in the linearized system is due to the

separation of the vorticity into the perturbed component and the ambient field. Because

of the interaction between these two parts, which is enabled by the term $\{h, \Psi_{0}\}$, the

perturbed component $\psi$ does not

describe aclosed

dynamical system.

The role of the

non-Hermitian

term [$w(x)\mathcal{G}\psi$ in (11)] is most simply

highlighted

by

Chandrasekhar’s model of ashear flow, which

assumes

apiece-wise linear flow $v(x)$ and

the corresponding delta

measure

$v’(x)[3]$. Before giving

amathematical

justification, let

us

examine formal solutions ofthis model.

In this subsection,

we

assume

$v(x)=0$ and consider

$i\partial_{t}\psi=w(x)\mathcal{G}\psi$ (20)

with

$w(x)=A\delta(x-a)$ $(A, a\in \mathrm{R})$. (21)

The formal eigenfunction of the generator, under the setting of $i\partial_{t}=\omega$ in (20), is

deter-mined by

$A \delta(x-a)\int_{-\infty}^{+\infty}G(x, \xi)\psi(\xi)d\xi=\omega\psi(x)$ , (22)

where $G(x, \xi)$ is the Green function of $\mathcal{G}$ [see (10)]. Solving (22),

we

obtain

$\omega$ $= \frac{A}{2}$, $\psi=\delta(x-a)$. (23)

We thus have

an

oscillation of a“surface wave” that is localized at $x=a$ and has the

wavenumber $k$ in the _$y$ direction [4].

If

we

have multiple “sources” of the surface waves, these

waves

interact through spatial

couplings induced by perturbed flows. Let

us

consider $N$ (finite number)

sources

$w(x)= \sum_{j=1}^{N}A_{j}\delta(x-a_{j})$ $(A_{j}, a_{j}\in \mathrm{R}, j=1, \ldots, N)$

.

(24)

The frequencies of the coupled surface

waves are

given by solving

$\sum_{j=1}^{N}A_{j}\delta(x-a_{j})\int_{-\infty}^{+\infty}G(x, \xi)\mathrm{s}\mathit{1})(\xi)d\xi=\omega\psi(x)$

.

(25)

Substitutin

$\psi=\sum_{j=1}^{N}\alpha_{j}\delta(x-a_{j})$,

into

(25),

we

obtain

the “dispersion

relation”

$\Lambda I$ $\{$ $\alpha_{1}$ . $\cdot$ . _– $\alpha_{N}/$ $=\omega$ $\{$ $\alpha_{1}$ . $\cdot$ . $\alpha_{N}/$ (26) with

$M_{i,j}=A_{i}G(a_{i}, a_{j})=A_{i} \frac{e^{-|a-a_{j}|}}{2}$

.

₍₂₇₎

The eigenvalue problem (26)

determines

the frequencies of the coupled

_{oscillations.}

Ob-viously, the matrix $M$ is non-symmetric (except for the

case

_{of $A_{j}=C$ for all}

$j$),

rep-resenting

the non-Hermitian

property of

the

generator.

For

some

sets of

coefficients

$A_{j}$

$(j=1, \cdots, N)$,

the frequency

$\omega$

can

be

complex.

The imaginary

part of$\omega$

gives the

growth

rate _{of the unstable mode of oscillation which corresponds to the}

_{“Kelvin-Helmholtz}

_in

ta-bility”.

3.3

Coupling

of the

two

generators

We

have

seen

the dynamics _{of vortices induced by each of the two different generators in}

(11), _{separately. Now,}

_we

_{study the coupling ofthese two generators.}

Let

us

first

consider the

case

of single

source;

see

$(2\mathrm{i})$. Theeigenvalue problem

associated

with the generalized Rayleigh equation (11) reads

$v(x) \psi+A\delta(x-a)\int_{-\infty}^{+\infty}G(x, \xi)\psi(\xi)d\xi=\omega\psi$, ₍₂₈₎

where

$G(x, \xi)=e^{-|x-\xi|}/2$ is the

Green function

_{[see (10)].}

_Let

_us

_{try to}

_{find aformal}

solution with assuming

$\psi=\alpha\delta(x-a)+\beta\delta(x-\mu)$, ₍₂₉₎

where

$\mu$ is

an

arbitrary

“fixed”

real number [see (13) and (23)].

Substituting

(29)

into (28.),

we obtain an eigenvalue problem

$L$ $(\begin{array}{l}\alpha\beta\end{array})=\omega$ $(\begin{array}{l}\alpha\beta\end{array})$

(30)

where

$L=\{$ $v(a)+AG(a, a)0$ $AG(a,\mu)v(\mu))$ . (31)

We

can

solve (30) to find aset ofeigenvalues and eigenfunctions:

$\omega$ $= \Omega_{1}(a):=v(a)+\frac{A}{2}$, $(\begin{array}{l}\alpha\beta\end{array})=U_{1}:=(\begin{array}{l}10\end{array})$ , (32)

and

$\omega$ $=\Omega_{c}(\mu):=v(\mu)$, _{$(\begin{array}{l}\alpha\beta\end{array})=U_{c}:=m(\mu)(\frac{AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),1},$}

$)$ , (33)

where the normalization factor is

$m( \mu)=[1+(\frac{AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a)})^{2}]-1/2$ (34)

The first eigenvalue$\Omega_{1}=v(a)+(A/2)$ gives the “Doppler-shifted” frequency of the surface

wave

[see (23)]. The corresponding formal eigenfunction is exactly $\psi=\delta(x-a)$ . The

second eigenvalue $\Omega_{c}=v(\mu)$ represents the local flow velocity [see (13)], while the

cor-responding formal eigenfunction describes acombination of the surface

wave

and alocal

vortex.

By the

transforms

$T=(U_{1}U_{c})=(01$ $\frac{m(\mu)AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),7n(\mu)},$ $)$ , $T^{-1}=(01$ $- \frac{AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),m(\mu)^{-1}},$ _$)$ , (35)

the matrix $L$ is diagonalized;

$T^{-1}LT=(\begin{array}{ll}\Omega_{1} 00 \Omega_{c}\end{array})$ .

We note that $T$ is not aunitary transform, reflecting the fact that the generator is not

a

Hermitian operator.

If the “resonance” $\Omega_{1}=\Omega_{c}[v(a)+A/2=v(\mu)]$ occurs, the second solution (33)

de-generates into the first

one

(32). This is the

case

when the matrix $L$ of (30) transforms

into aJordan block. We introduce ageneralized eigenfunction belongingto the degenerate

eigenvalue $\Omega_{1}$;

$U_{c}’=(\begin{array}{l}1(AG(a,\mu))^{-1}\end{array})$ , (36)

which satisfies $(L-\Omega_{1}I)^{2}U_{c}’=0$. By transforms

$T’=(U_{1}U_{c}’)=(\begin{array}{ll}1 10 (\Lambda G(a,l^{l}))^{-1}\end{array})$ , $T^{\prime-1}=(\begin{array}{lll}1 -AG(a \mu)0 AG(a,\mu) \end{array})$ ,

we

can

transform $L$ into

aJordan

canonical form

$T^{\prime-1}LT’=(\begin{array}{ll}\Omega_{1} 10 \Omega_{1}\end{array})$ .

To unify both the non-resonant and resonant (nilpotent) cases,

we

define

$\tilde{m}(\mu)=\{$

$m(\mu)$ if $\Omega_{c}(\mu)\neq\Omega_{1}(a)$

$(AG(a, \mu))^{-1}$ if $\Omega_{c}(\mu)=\Omega_{1}(a)$ (i.e. $m(\mu)=0$), (37)

and combine $U_{c}$ and $U_{c}’$

as

$\tilde{U}_{c}(\mu)=(\frac{m(\mu)AG(a,\mu)}{\Omega_{\mathrm{c}}(l^{l})-\Omega_{1}(a),\tilde{m}(\mu)},$ $)$ . (38)

The transform

$\tilde{T}=(U_{1}\tilde{U}_{c}(\mu))=(01$ $\frac{n\iota(\mu)AG\prime(a,\mu)}{\Omega_{r}(\mu)-\Omega_{1}(a),\tilde{m}(\mu)},$ $)$ (39)

is regular for all $\mu$.

Next,

we

study the

case

ofmultiple sources;

see

(24). We solve

$v(x) \psi+\sum_{j=1}^{N}A_{j}\delta(x-a_{j})\int_{-\infty}^{+\infty}G(x, \xi)\psi(\xi)d\xi=\omega\psi$ (40)

with assuming

$\psi=\sum_{j=1}^{N}\alpha_{j}\delta(x-a_{j})+\beta\delta(x-\mu)$.

To generalize the above calculations,

we

prepare notation [see (32)]

$\Omega_{j}(a_{j})=v(a_{j})+\frac{A_{j}}{2}$ $(j=1, \ldots, N)$. (41)

The dispersion relation is

$L$ $(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\beta\end{array})=\omega$ $(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\beta\end{array})$ . (42)

where the matrix $L$ generalizes (31)

as

$L=\{$ $\mathrm{f}l_{1}(..\cdot a_{1})0^{\cdot}.$ . $A_{1}G(a_{1}.\cdot.’ a_{N})\Omega_{N}(a_{N})0$ $A_{N}G(..\cdot a_{N},\mu)A_{1}G(a_{1},\mu)\Omega_{c}(\mu))$ . $A_{N}G(a_{r\mathrm{V}}, a_{1})$

(43)

We have two

different

classes of solutions. The first

group,

corresponding to (32), is

ob-tained with setting $\beta$ $=0$

.

Then, the eigenvalue problem (42) reduces into

$(\begin{array}{lll}\Omega_{1}(a_{1})\ddots A_{1}G(a_{1},a_{N})\vdots \ddots \vdots A_{N}G(a_{N},a_{1}) \Omega_{N}(a_{N})\end{array})(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\end{array})=\omega$$(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\end{array})$ , (44)

which reads

as

the dispersion relation that is Doppler shifted from (26). The second class

ofeigenvectors, corresponding to (33), is given by setting $\beta\neq 0$. The eigenvalue is

$\Omega_{c}(\mu)=v(\mu)$,

and the corresponding eigenfunction is determined by

$(\begin{array}{lll}\Omega_{1}(a_{1})-\Omega_{c}(\mu) A_{1}G(a_{1},a_{N})\vdots \ddots \vdots A_{N}G(a_{N},a_{\mathrm{l}}) \Omega_{N}(a_{N})-\Omega_{c}(\mu)\end{array})(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\end{array})=-\beta$ $(\begin{array}{l}A_{1}G(a_{1},\mu)\vdots A_{N}G(a_{N},\mu)\end{array})$

.

(45)

As discussed above, the

resonances

$\Omega_{j}(a_{j})=\Omega_{c}(\mu)(j=1, \cdots, N)$ yield singularities in

the

matrix

of

(45),

and

then,

we

must

consider

nilpotents

4

Spectral

resolution

of coupled

non-Hermitian

gen-erator

In this section,

we

formulate the vortex dynamics equation (11) with the delta-measure

field (24)

as an

evolution equation in

an

appropriate Hilbert

space,

and give aspectral

resolution of the generator. The generator reads

$\mathcal{L}\psi=v(x)\psi+\sum_{j=1}^{N}A_{j}\delta(x-a_{j})\int_{-\infty}^{+\infty}G(x, \xi)\psi(\xi)d\xi$, (46)

where $v(x)\in C(\mathrm{R})$, $A_{j}\in \mathrm{R}$, $a_{j}\in \mathrm{R}(j=1, \ldots, N)$, and $G(x, \xi)=e^{-|x-\xi|}/2$ is the

Green

function [see (10)]. In what follows,

we

assume

$|v(x)|<c(\forall x)$ with

some

finite number $c$

.

Since the delta

measure

$\delta(x-a_{j})$ is not amember ofthe Lebesgue space,

we

encounter

adifficulty in formulating the problem in the

conventional

$L^{2}$ Hilbert

space.

4,1

Mathematical formulation of

the

generator

Let

us

consider

aHilbert space

$V=\mathrm{C}^{N}\oplus L^{2}(\mathrm{R})$, (47)

where $\mathrm{C}^{N}$ is the unitary space of dimension $N$, and $L^{2}(\mathrm{R})$ is the complex Lebesgue

space

on

$\mathrm{R}$

endowed

with the

standard

inner product. The member of$V$ is written

as

$\psi=(\varphi(x)\alpha)$ $[\alpha\in \mathrm{C}^{N}, \varphi(x)\in L^{2}(\mathrm{R})]$. (48)

The inner product of $V$ is, thus,

defined as

$\langle\psi, \psi’\rangle=(\alpha, \alpha’)+(\varphi, \varphi’)=\sum_{j=1}^{N}\alpha_{j}\overline{\alpha}_{j}’+\int_{-\infty}^{+\infty}\varphi(x)\overline{\varphi}’(x)dx$ (49)

We identify

$\uparrow\int J=(\varphi(x)\alpha)\Leftrightarrow\psi(x)=\sum_{j=1}^{N}\alpha_{j}\delta(x-a_{j})+\varphi(x)$. (50)

It isessential todecomposethedelta-measure part (representingthesurfacewaves) fromthe

total vorticity $\psi$. Although the supports (in the

sense

of

distributions) of both components

$\delta(x-a_{j})$ and $\varphi(x)$ may overlap,

we

separate them into

different

degrees

of

freedom.

Because $\mathcal{G}\psi\in C(\mathrm{R})$ for all $\psi\in V$, the generator $\mathcal{L}$ is

abounded

operator

on

$V$.

Following (50), the generator $\mathcal{L}$ of (46) is

now

written in

amatrix

form [see (43)]

$\mathcal{L}\psi=\{$ $\Omega_{1}(...a_{1})0^{\cdot}.$. $A_{1}G(a_{1}.\cdot.’ a_{N})0$ $\int A_{N}G(a_{N}.\cdot.,x)\cdot dx\int A_{1}G(a_{1},x)\cdot dx\Omega_{c}(x))$

$(\begin{array}{l}\alpha_{1}\vdots\alpha_{N}\varphi(x)\end{array})$

$A_{N}G(a_{N}, a_{1})$ $\Omega_{N}(a_{N})$

(51)

In the previous section,

we

dealt delta functions in aformal way and did calculations

using $\delta(x-\mu)$ with

an

arbitrary $\mu\in \mathrm{R}$ [see (13) and (29)]. We note that such formal

functions

are

not the member

of

the Hilbert

space

$V$. In this section, they

are

justified

as

generalized

eigenfunctions corresponding to “continuous spectr\"a”.

4.2

Spectral

resolution of

the generator

First,

we

consider the simple

case

ofsingle “source”, i.e., $w(x)=A\delta(x-a)$ [see (21)]. The

surface

wave

mode has only

one

degree offreedom $(N=1)$. Here, the generator $\mathcal{L}$ of (51)

simplifies

as

$\mathcal{L}=($ $\Omega_{1}(a)0$ _{$\int AG(a,x)\Omega_{c}(x)$}

.

$dx$

).

(52)

As

we

have shown in Sec. 3.3, there

are

two different classes offormal eigenfunctions [see

(32) and (33)$]$; In the form consistent to the notation of(48), they read

$\Omega_{1}(a)=v(a)+\frac{A}{2}$, $U_{1}=(\begin{array}{l}\mathrm{l}0\end{array})$ $(53\grave{J}$

$\Omega_{c}(\mu)=v(\mu)$, $\tilde{U}_{c}(\mu)=(\tilde{m}’\mu)\frac{m(\mu)AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),(\mu)\delta(x-})$ . (54)

The first eigenfunction represents the surface

wave.

The second

one

includes

an

arbitrary

real number $\mu$, correspondingto the continuous spectrum, and asingular

function

_{$\delta(x-\mu)$}.

We must

integrate

(54)

over

$\mu\in \mathrm{R}$ to span the complete basis of $V$

.

Formally,

we

can

generalize the transform $\tilde{T}$

of(39)

as

$\mathcal{T}=(U_{1}\int(\cdot, \delta(x-\mu))\tilde{U}_{c}(\mu)d\mu)=(01$ $\int(\cdot,\delta(x-\mu))\tilde{m}’\mu)d\mu\int(\cdot,\delta(x-\mu))\frac{m(\mu)AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),(\mu)\delta(x-}d\mu)$. (55)

To cast this

formal

expression in

an

appropriate

mathematical

representation,

we

invoke

the

resolution

of the identity (17). The

formal

correspondence is

$\int_{-\infty}^{+\infty}(u(x), \delta(x-\mu))\delta(x-\mu)d\mu=\int_{-\infty}^{+\infty}$ dE(\mu )u_$=u$.

We also

define

$F( \mu)u=\int_{-\infty}^{\mu}u(x)dx$, (56)

which gives

$dF(\mu)u=u(\mu)d\mu$.

Using this notation,

we can

write

$\int f(\mu)dF(\mu)u(x)=\int f(\mu)u(\mu)d\mu=\int f(x)u(x)dx$.

The operator $\mathcal{T}$ is

now

written in arigorous form of

$\mathcal{T}=($ $01$ $\int\frac{m(\mu)AG(a,\mu)}{\Omega_{c}(\mu)-\Omega_{1}(a),\int\tilde{m}(\mu)dE},dF(\mu)(\mu))=(01$ $\int\frac{m(x)AG(a,x)}{\Omega_{c}(x)-\Omega_{1}(a),\tilde{m}(x)},$ _{$\cdot dx)$} (57)

Reflecting the non-Hermitian propertyof thegenerator$\mathcal{L}$, theoperator$\mathcal{T}$isnot aunitary

transform. By combing both non-resonant and resonant (nilpotent)

cases

[cf. (39)], this $\mathcal{T}$

is aregular transform. The inverse operator is

$\mathcal{T}^{-1}=($ $01$ $- \int(\frac{m(x)}{\overline{m}(x)})(\frac{AG(a,x)}{\Omega_{c}(x)-\Omega_{1}(a),(x)^{-1}},)\tilde{m}$

.

$dx$

).

(581

Using the transforms $\mathcal{T}$ and $\mathcal{T}^{-1}$,

we

obtain the Jordan canonical form of$\mathcal{L}$;

$\mathcal{T}^{-1}\mathcal{L}\mathcal{T}=$ _{$(\Omega_{1}0$} $\int\Omega_{c}(\mu)dE(\mu)\int\rho(\mu)dF(\mu))$

$=$ $(\begin{array}{llll}\Omega_{1} /\backslash \rho(x)\cdot dx0 \Omega_{c}(x) \end{array})$ , (59)

where

$\rho(x)=\{$ 1if

$\Omega_{c}(\mu)=\Omega_{1}(a)$

0if

$\Omega_{c}(\mu)\neq\Omega_{1}(a)$

The support of$\rho(x)$

can

have afinite

measure

when the

resonance

condition $\Omega_{c}(\mu)=\Omega_{1}(a)$

holds

on a

finite interval of$x$.

4.3

Spectral

representation

of the

propagator

The propagator $e^{-it\mathcal{L}}$ is defined by solving the initial value problem for (11)

$\{$

$i\partial_{t}\psi=\mathcal{L}\psi$,

$\psi(0)=\psi_{0}$

(60)

and writing the solution

as

$(/)(t)=e^{-it\mathcal{L}}\psi_{J_{0}}$.

Defining $\psi=\mathcal{T}\chi$,

we

transform (60) into

$\{$

$i\partial_{t}\chi=\mathcal{T}^{-1}\mathcal{L}\mathcal{T}\chi$,

$\chi(0)=\mathcal{T}^{-1}\psi_{0}$.

(61)

Using the spectral resolution (59), the solution of (61) is given by

$e^{-it\mathcal{T}^{-1}\mathcal{L}\mathcal{T}}$

$=$ $(c_{0}^{\mathrm{J}}-it\Omega_{1}$ $- \int ite^{-it\Omega_{1}}\rho(\mu)dF(\mu)\int e^{-il\Omega_{c}(\mu)}dE(\mu))$

$=$

(

$e_{0}^{-it\Omega_{1}}$ -/

$\cdot$

$ite^{-it\Omega_{1}},\rho(x)e^{-it\Omega_{c}(x)}\cdot d.c$

).

(62)

The solution of (60) is given by

$\psi(t)=\mathcal{T}[e^{-it\mathcal{T}^{-1}\mathcal{L}\mathcal{T}}]\mathcal{T}^{-1},\psi_{0}$ .

Using (57) and (58), we obtain

$e^{-it\mathcal{L}}$ _$=$ $\mathcal{T}$

(

$e_{0}^{-it\Omega_{1}}$ $- \int ite^{-it\Omega_{1}},\rho(x)e^{-it\mathrm{f}\mathit{1}_{c}(x)}\cdot dx$

)

$\mathcal{T}^{-1}$

$=$ $(\begin{array}{ll}e^{-il\Omega_{1}} \lrcorner \mathrm{Y}0 e^{-it\Omega_{c}(x)}\end{array})$ , (63)

[e $\ovalbox{\tt\small REJECT} t\mathrm{O}_{c}(x)$

e

$\ovalbox{\tt\small REJECT} t0_{1}(a)]_{\ovalbox{\tt\small REJECT}^{\ovalbox{\tt\small REJECT}}}1\mathrm{C}\mathrm{I}\ovalbox{\tt\small REJECT}(a,$r)

$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}}\ldots$

$\yen$

ile $AG(a_{\ovalbox{\tt\small REJECT}}x)p(x)$

.

dx,

$\mathrm{O}_{c}(\ovalbox{\tt\small REJECT} \mathrm{r})-\mathrm{O}.(a)$

and

we

have used the relations

$\{$

$\frac{m(x)}{\tilde{m}(x)}=1-\rho(x)$

$\frac{\rho(x)}{\tilde{m}(x)}=AG(a, x)\rho(x)$

The off-diagonal part $X$ of the matrix operator (63) represents the mode interactions

originating from the

non-Hermitian

propertyof the generator. The $X$ consists oftwo parts;

one

is the contribution from the non-resonant flow in the region of the support of$1-\rho(x)$,

andthe otheris from theresonant flow inthat of$l^{y}(x)$. Thelatterproducessecular behavior

(represented by the factor $ite^{-it\Omega_{1}}$).

References

[1] K. Yosida, Functional Analysis (Springer-Verlag, Berlin, 1995).

[2] By changing

the scale

of y,

we

can

normalize k

to 1.

On the

contrary, to

take k

$\neq 1$,

we translate y $arrow ky$,

v

$arrow kv$, w $arrow kw$ and $e^{-|x-\xi|}/2arrow e^{-k|x-\xi|}/(2k)$ in the later

calculations;

see

(8), (11) and (9).

[3]

S.

Chandrasekhar, Hydrodynamic and hydromagnetic stability (Clarendon, Oxford,

1961).

[4] There

are

rich examples of relevant phenomena; The Rossby

waves

of perturbations

in geological jet streams, the diocotron

waves

in non-neutral plasmas, and the drift

waves

in neutral plasmas